Difference between revisions of "2009 AIME I Problems/Problem 1"
m |
|||
| Line 15: | Line 15: | ||
For the maximum, we have that <math>b^2 = 9c</math>; <math>b</math> is maximized when <math>9c</math> is the greatest possible perfect square; this happens when <math>c = 4</math>, yielding <math>b = 6</math>. Thus, the largest possible geometric number is 964. | For the maximum, we have that <math>b^2 = 9c</math>; <math>b</math> is maximized when <math>9c</math> is the greatest possible perfect square; this happens when <math>c = 4</math>, yielding <math>b = 6</math>. Thus, the largest possible geometric number is 964. | ||
| − | Our answer is thus <math>964 - 124 = | + | Our answer is thus <math>964 - 124 = 840. |
==Solution 3== | ==Solution 3== | ||
| − | The smallest geometric number is 124 because 123 and any number containing a zero does not work. 964 is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives <math>\boxed{840}. | + | The smallest geometric number is 124 because 123 and any number containing a zero does not work. 964 is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives </math>\boxed{840}.$ |
==Video Solution== | ==Video Solution== | ||
Revision as of 20:10, 11 May 2020
Problem
Call a 
-digit number geometric if it has distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
Solution 1
Assume that the largest geometric number starts with a nine. We know that the common ratio must be a rational of the form 
for some integer 
, because a whole number should be attained for the 3rd term as well. When 
, the number is 
. When 
, the number is 
. When 
, we get 
, but the integers must be distinct. By the same logic, the smallest geometric number is 
. The largest geometric number is and the smallest is . Thus the difference is 
.
Solution 2
Consider the three-digit number 
. If its digits form a geometric sequence, we must have that 
, that is, 
.
The minimum and maximum geometric numbers occur when 
is minimized and maximized, respectively. The minimum occurs when 
; letting 
and 
achieves this, so the smallest possible geometric number is 124.
For the maximum, we have that 
; 
is maximized when 
is the greatest possible perfect square; this happens when , yielding 
. Thus, the largest possible geometric number is 964.
Our answer is thus $964 - 124 = 840.
==Solution 3== The smallest geometric number is 124 because 123 and any number containing a zero does not work. 964 is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives$ (Error compiling LaTeX. Unknown error_msg)\boxed{840}.$
Video Solution
~IceMatrix
See also
| 2009 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
